Why Prefix Sum Array in DSA Java?

The Big Idea

What if you could get the total of any part of a list instantly without adding every time?

The Scenario

Imagine you have a long list of daily sales numbers, and you want to quickly find the total sales between any two days. Doing this by adding each day's sales every time you ask is like counting coins one by one every time you want to know your total money.

The Problem

Manually adding numbers each time is slow and tiring. If the list is long, it takes a lot of time and you might make mistakes while adding. It's like repeatedly counting the same coins again and again.

The Solution

A prefix sum array keeps a running total of the sales up to each day. This way, you can find the total sales between any two days by just subtracting two numbers, making it very fast and easy.

Before vs After

✗ Before

int sum = 0;
for (int i = start; i <= end; i++) {
    sum += sales[i];
}

✓ After

int total = prefixSum[end + 1] - prefixSum[start];

What It Enables

This lets you answer sum queries instantly, even on huge lists, saving time and effort.

Real Life Example

Online stores use prefix sums to quickly calculate total sales in any date range, helping them track performance without delay.

Key Takeaways

Manually summing ranges is slow and error-prone.

Prefix sum arrays store running totals to speed up queries.

They make repeated sum calculations fast and simple.

Practice

1. What does a prefix sum array represent for a given array arr?

easy

A. An array where each element is the sum of all elements up to that index in arr

B. An array where each element is the product of all elements up to that index in arr

C. An array where each element is the difference between consecutive elements in arr

D. An array where each element is the maximum value found up to that index in arr

Solution

1. Step 1: Understand prefix sum definition

   A prefix sum array stores sums of elements from the start of the array up to each index.

2. Step 2: Compare options with definition

   Only An array where each element is the sum of all elements up to that index in arr matches this definition exactly.

3. Final Answer:

   An array where each element is the sum of all elements up to that index in arr -> Option A

4. Quick Check:

   Prefix sum = cumulative sum [OK]

Hint: Prefix sum means cumulative sum from start to current index [OK]

Common Mistakes:

• Confusing prefix sum with product or max operations
• Thinking prefix sum stores differences
• Mixing prefix sum with suffix sum

2. Which of the following Java code snippets correctly initializes a prefix sum array prefix for an integer array arr?

easy

A. int[] prefix = new int[arr.length]; for(int i=0; i

B. int[] prefix = new int[arr.length]; prefix[0] = 0; for(int i=1; i

C. int[] prefix = new int[arr.length]; prefix[0] = arr[0]; for(int i=1; i

D. int[] prefix = new int[arr.length]; for(int i=0; i

Solution

1. Step 1: Check initialization of prefix array

   Prefix sum array must start with prefix[0] = arr[0] to include first element sum.

2. Step 2: Verify loop calculation

   Loop from i=1 to end adds previous prefix sum plus current arr element, which is correct.

3. Final Answer:

   int[] prefix = new int[arr.length]; prefix[0] = arr[0]; for(int i=1; i -> Option C

4. Quick Check:

   Initialize first element, then add previous sums [OK]

Hint: Start prefix[0] with arr[0], then add previous sums in loop [OK]

Common Mistakes:

• Starting prefix[0] with 0 instead of arr[0]
• Multiplying instead of adding elements
• Reversing array elements instead of summing

3. What is the output of the following Java code?

int[] arr = {2, 4, 6, 8};
int[] prefix = new int[arr.length];
prefix[0] = arr[0];
for(int i = 1; i < arr.length; i++) {
    prefix[i] = prefix[i-1] + arr[i];
}
System.out.println(java.util.Arrays.toString(prefix));

medium

A. [2, 6, 12, 20]

B. [2, 4, 6, 8]

C. [0, 2, 6, 12]

D. [2, 8, 14, 22]

Solution

1. Step 1: Calculate prefix sums step-by-step

   prefix[0] = 2; prefix[1] = 2 + 4 = 6; prefix[2] = 6 + 6 = 12; prefix[3] = 12 + 8 = 20;

2. Step 2: Verify final prefix array

   The prefix array is [2, 6, 12, 20].

3. Final Answer:

   [2, 6, 12, 20] -> Option A

4. Quick Check:

   Sum accumulates: 2, 6, 12, 20 [OK]

Hint: Add current element to previous prefix sum for each index [OK]

Common Mistakes:

• Printing original array instead of prefix sum
• Starting prefix sum with 0 incorrectly
• Adding wrong elements in loop

4. Identify the error in the following Java code that attempts to create a prefix sum array:

int[] arr = {1, 3, 5};
int[] prefix = new int[arr.length];
prefix[0] = 0;
for(int i = 1; i <= arr.length; i++) {
    prefix[i] = prefix[i-1] + arr[i];
}

medium

A. Loop condition should be i < arr.length instead of i <= arr.length

B. prefix[0] should be initialized to arr[0] instead of 0

C. Array index arr[i] will cause an error when i equals arr.length

D. All of the above

Solution

1. Step 1: Check loop boundary

   The loop uses i <= arr.length, which causes index out of bounds because valid indices are 0 to arr.length-1.

2. Step 2: Check prefix[0] initialization and array access

   prefix[0] should be arr[0] to start sums correctly; also arr[i] access fails at i = arr.length.

3. Step 3: Combine all errors

   All issues cause runtime errors or incorrect prefix sums.

4. Final Answer:

   All of the above -> Option D

5. Quick Check:

   Loop bounds, initialization, and indexing errors combined [OK]

Hint: Check loop limits and array indices carefully [OK]

Common Mistakes:

• Using <= instead of < in loop condition
• Initializing prefix[0] incorrectly
• Accessing array out of bounds

5. Given an integer array arr of length 5: {3, 1, 4, 1, 5}, you want to find the sum of elements from index 1 to 3 (inclusive) efficiently using a prefix sum array. Which expression correctly calculates this sum using the prefix sum array prefix where prefix[i] is the sum of elements from index 0 to i?

hard

A. prefix[3] - prefix[2]

B. prefix[3] - prefix[0]

C. prefix[3] - prefix[1]

D. prefix[3] - prefix[0] + arr[1]

Solution

1. Step 1: Understand prefix sum usage for range sums

   Sum from index l to r = prefix[r] - prefix[l-1]. Here l=1, r=3.

2. Step 2: Apply formula with given indices

   Sum = prefix[3] - prefix[0]. prefix[0] is sum up to index 0, so subtracting it leaves sum from 1 to 3.

3. Final Answer:

   prefix[3] - prefix[0] -> Option B

4. Quick Check:

   Range sum = prefix[r] - prefix[l-1] [OK]

Hint: Range sum = prefix[end] - prefix[start - 1] [OK]

Common Mistakes:

• Subtracting prefix at wrong index
• Adding extra elements unnecessarily
• Confusing inclusive indices